 Structure of Complex Atoms - Continued

Quantum Mechanics – Molecular Structure

Jeong Won Kang

Department of Chemical Engineering Korea University

Applied Statistical Mechanics Lecture Note - 4

Subjects

 Structure of Complex Atoms - Continued

 Molecular Structure

Structure of Complex Atoms

- Continued

The Spectra of Complex Atoms

 The spectra of atoms rapidly become very

complicated as the number of electrons increases.

 Spectra of an atom : The atom undergoes transition with a change of energy; ΔE = hν

Gives information about the energies of electron

However, the actual energy levels are not given solely by the energies of orbitals

Electron-Electron interactions

Measuring Ionization Energy (H atoms)

 The plot of wave numbers vs. 1/n² gives the ionization energy for hydrogen atoms

2 2

n R hc

I E I

hc E n

R c

h E

E E

H lower

lower H

lower

−

=

=

−

−

=

=

=

−

= Δ

ν ν ν

ν

Slope = I

Quantum defects and ionization energies

 Energy levels of many-electron atoms do not vary as 1/n²

 The outermost electron

 Experience slightly more charge than 1e

 Other Z-1 atoms cancel the charge slightly lower than 1

 Quantum Defect (δ): empirical quantity

2 2 2

) (

n R hc

I n E hcR

n E hcR^H

−

=

− −

=

−

=

ν

δ

Rydberg state

Pauli exclusion principle and the spins

 Palui exclusion principle

“ No more than two electrons may occupy any given orbital and, if two occupy one orbital, then their spin must be paired “

“ When label of any two identical fermions are

exchanged, the total wavefunction changes sign. When the label of any two identical bosons are exchanged, the total wavefunction retain its sign “

Pauli exclusion principle and the spins

 Two electrons (fermions) occupy an orbital Ψ then ;

 Total Wave function = (orbital wave function)*(spin wave function) )

1 , 2 ( )

2 , 1

( = −Ψ Ψ

) 2 ( ) 1 ( α α

) 2 ( ) 1 ( ψ ψ

) 2 ( ) 1 ( β

α α⁽²⁾β⁽¹⁾ β⁽¹⁾β⁽²⁾ four possibility

{ }

{ ^{(1) (2) (1) (2)}}

) 2 / 1 ( ) 2 , 1 (

) 2 ( ) 1 ( )

2 ( ) 1 ( ) 2 / 1 ( ) 2 , 1 (

2 / 1

2 / 1

α β β

α σ

α β β

α σ

−

=

+

=

− +

We cannot tell which one is α and β

Normalized Linear Combination of two spin wave functions

) 2 ( ) 1 ( ) 2 ( ) 1 (

) 2 , 1 ( ) 2 ( ) 1 (

) 2 , 1 ( ) 2 ( ) 1 (

) 2 ( ) 1 ( ) 2 ( ) 1 (

β β ψ

ψ

σ ψ

ψ

σ ψ

ψ

α α ψ

ψ

− +

Pauli exclusion principle and the spins

) 2 ( ) 1 ( ) 2 ( ) 1 (

) 2 , 1 ( ) 2 ( ) 1 (

) 2 , 1 ( ) 2 ( ) 1 (

) 2 ( ) 1 ( ) 2 ( ) 1 (

β β ψ

ψ

σ ψ

ψ

σ ψ

ψ

α α ψ

ψ

− +

) 1 , 2 ( )

2 , 1

( = −Ψ Ψ

{ }

{ ^{(1) (2) (1) (2)}} ^(2,1)

) 2 / 1 ( ) 2 , 1 (

) 1 , 2 ( )

2 ( ) 1 ( )

2 ( ) 1 ( ) 2 / 1 ( ) 2 , 1 (

2 / 1

2 / 1

−

−

+ +

−

=

−

=

= +

=

σ α

β β

α σ

σ α

β β

α σ

Requirement :

) 1 , 2 ( )

2 , 1 (

) 1 , 2 ( )

2 , 1 (

) 1 , 2 ( )

2 , 1 (

) 1 , 2 ( )

2 , 1 (

Ψ

= Ψ

Ψ

−

= Ψ

Ψ

= Ψ

Ψ

= Ψ

) 2 , 1 ( ) 2 ( ) 1

( ψ σ₋

ψ Only one acceptable

Singlet and Triplet State

 Excited State of He atom

 1s²  1s¹ 2s¹

 The two electrons need not to be paired

 Singlet : paired spin arrangement

 Triplet : parallel spin arrangement

 Hund’s principle : triplet states generally lie lower than triplet state )

2 ( ) 1 ( α α

) 2 ( ) 1 ( β β

{ ^{(1) (2) (1) (2)}}

) 2 / 1 ( ) 2 , 1

( ^1/2 α β β α

σ₊ = +

{ ^{(1) (2) (1) (2)}}

) 2 / 1 ( ) 2 , 1

( ^1/2 α β β α

σ₋ = −

Spectrum of atomic Helium

 Spectrum of He atom is more complicated than H atom

 Only one electron is excited

• Excitation of two electrons require more energy than ionization energy

 No transitions take place between singlet and triplet states

• Behave like two species

Spin-Orbit Coupling

 Electron spins has a further implication for energies of atoms when l > 0 ( finite orbital angular momentum )

 (spin magnetic momentum) + (magnetic moment doe to orbital angular momentum)  spin-orbit coupling

2 + 1

= l j

Parallel  high angular momentum

Opposed  low angular momentum

2

− 1

= l j

Spin-orbit coupling

 When l=2

 Spin-Orbit Coupling constant (A)

 Dependence of spin-orbit interaction on the value of j

2 3 2 1

2 5 2 1

=

−

=

= +

= l j

l j

)) 1 ( ) 1 ( ) 1 ( 2 (

1

,

, = hcA j j+ −l l + −s s+

E_{l j s}

Spin-Orbit Coupling

 Spin-orbit coupling depends on the nuclear charge

 The greater the nucleus charge  the stronger spin-orbit coupling

 Very small in H , very large in Pb

 Fine structure

 Two spectral lines are observed

 The structure in a spectrum due to spin- orbit coupling

 Ex) Na (street light)

Term symbols and selection rules

 Skip…

Molecular Structure

Topics

 Valence-Bond Theory

 Molecular Orbital Theory

Born-Oppenheimer approximation

 Assumption

 The nuclei is fixed at arbitrary location

 H2 molecule

• Nuclei move about 1 pm

• Electrons move about 1000 pm

 Use

 different separation

solve Schrodinger equation

Energy of molecules vary with bond length

 Equilibrium bond length (R_e)

 Bond Dissociation energy (D_e)

Structure Prediction, Property Estimation

Valence-Bond Theory

 Consider H₂ molecule

 If electron 1 is on atom A and electron 2 is on atom B

) ( )

( _{1 1 1}

1 r r

A

A H S

S

H ψ

ψ ψ =

) 2 ( ) 1 ( B

= A

ψ ψ = ^A(2)B(1) ) 1 ( ) 2 ( )

2 ( ) 1

( B A B

A ±

ψ =

) 1 ( ) 2 ( )

2 ( ) 1

( B A B

A +

ψ =

Simple notation

Linear combination of wave functions

Lower energy

σ - bond

 Cylindrical symmetry around internuclear axis

 Rambles s-orbital : called sigma-bond

 Zero angular momentum around internuclear axis

 Molecular potential energy

 Spin : spin paring

 According to Pauli principle spin must be paired

π-bond

 Essence of valence-bond theory

 Pairing of the electrons

 Accumulation of electron density in the internulear region

 N2 atom

1

px 1

py

1 1 1

22 2 2

2s p_x p_y p_z

1

pz σ−bond

π−bond

Cannot form σ-bond not symmetrical around internuclear axis

Water molecule

1 1

2

22 2 2

2s p_x p_y p_z

Bond angle : 90 degree

 Experimental 104.5 degree ! Two σ−bonds

Promotion

 CH₄ molecule

 cannot be explained by valence-bond theory

 only two electrons can form bonds

 Promotion

 Excitation of electrons to higher energy

 If bonds are formed (lower energy) , excitation is worthwhile

 4 σ-bonds

1 1

22 2

2s p_x p_y

1 1 1

12 2 2

2s p_x p_y p_z

Hybridization

 Description of methane is still incomplete

 Three s-bond (H1s – C2p) + one s-bond (H1s – C2s)

 Cannot explain symmetrical feature of methane molecules

 Hybrid orbital

 Interference between C2s + C2p orbital

 Forming sp³ hybrid orbital

z y

x

z y

x

z y

x

z y

x

p p

p s h

p p

p s h

p p

p s h

p p

p s h

−

− +

=

− +

−

=

+

−

−

=

+ +

+

=

4 3 2 1

Hybridization

 Formation of σ-bond in methane

) 1 ( ) 2 ( )

2 ( ) 1

( ₁

1 A h A

h +

ψ =

sp ² hybridization : ethylene

 sp² hybridization

y x

y x

x

p p

s h

p p

s h

p s

h

2 / 1 2

/ 1 3

2 / 1 2

/ 1 2

2 / 1 1

2) (3 2)

(3

2) (3 2)

(3 2

−

−

=

− +

= +

=

sp hybridization : acetylene

 sp hybridization

z z

p s h

p s h

−

= +

=

2 1

Molecular Orbital Theory

 Molecular Orbital (MO) Theory

 Electrons should be treated as spreading throughout the entire molecule

 The theory has been fully developed than VB theory

 Approach :

Simplest molecule ( H₂⁺ ion)  complex molecules

The hydrogen molecule-ion

 Hamiltonian of single electron in H₂⁺

 Solution : one-electron wavefunction

 Molecular Orbital (MO)

 The solution is very complicated function

 This solution cannot be extended to polyatomic molecules

1 ) 1

( 1 4

2 _{0 1 1}

2 2

1 r r R

V e m V

H

B A

e

− +

−

= +

∇

−

= ^ πε

Attractions between electron and nuclei Repulsion between the nuclei

Linear Combination of Atomic Orbitals (LCAO-MO)

 If an electron can be found in an atomic orbital belonging to atom A and also in an atomic orbital belonging to atom b, then overall

wavefuntion is superposition of two atomic orbital :

 N : Normalization factor

 Called a σ-orbital

( ) ( )

/ 1

2 / 3 1 0 / 1

0 0

) (

a B e

a A e

B A N

a r S

H

a r S

H

B

B

A

A

ψ π ψ π ψ

−

−

±

=

=

=

=

±

= Linear Combination of

Atomic Orbitals (LCO-MO)

Normalization

( ) ( )

/ 1

2 / 3 1 0 / 1

0 0

) (

a B e

a A e

B A N

a r S

H

a r S

H

B

B

A

A

ψ π ψ π ψ

−

−

±

=

=

=

=

±

=

{ }

56 . 0

59 . 0

) 2 1 1 ( 2

1

2 2

2 2

*

*

=

≈

=

+ +

= +

+

=

=













N

d AB S

S N

d AB d

B d

A N

d d

τ

τ τ

τ τ

ψ ψ

τ ψ ψ

{ }

10 . 1

59 . 0

) 2 1 1 ( 2

1

2 2

2 2

*

*

=

≈

=

− +

=

− +

=

=













N

d AB S

S N

d AB d

B d

A N

d d

τ

τ τ

τ τ

ψ ψ

τ ψ ψ ψ+

ψ−

LCAO-MO

Amplitude of the bonding orbital in hydrogen molecule-ion

r_A and r_Bare not independent

{

}

r_B = _A + − _A

Bonding Orbital

 Overlap density  Crucial term

 Electrons accumulates in the region where atomic orbital overlap and interfere constructively.

) 2 ( ^{2 2}

2

2 = N A +B + AB

ψ+

Probability density if the electron were confined to the atomic orbital A

Probability density if the electron were confined to the atomic orbital B

An extra contribution to the density (Overlap density)

Bonding Orbital

 The accumulation of electron density between the nuclei put the electron in a position where it interacts strongly with both nuclei

 the energy of the molecule is lower than that of separate atoms

Bonding Orbital

 Bonding orbital

 called 1σ ^orbital

 σ ^electron

 The energy of 1σ orbital decreases as R decreases

 However at small separation, repulsion becomes large

 There is a minimum in potential energy curve

 Re = 130 pm (exp. 106 pm)

 De = 1.77 eV (exp. 2.6 eV)

Antibonding Orbital

 Linear combination ψ_- corresponds to a higher energy

 Reduction in probability density between the nuclei (-2AB term)

 Called 2σ orbital (often labeled 2σ ^*)

) 2 ( ^{2 2}

2

2 = N A +B − AB

ψ−

Amplitude of antibonding orbital

Antibonding orbital

 The electron is excluded from internuclear region

 destabilizing

 The antibonding orbital is more antibonding than the bonding orbital is

bonding

s H s

H E E

E

E₋ − ₁ > ₊ − ₁

Molecular orbital energy diagram

The Structure of Diatomic Molecules

 Target : Many-electron diatomic molecules

 Similar procedure

 Use H₂⁺ molecular orbital as the prototype

 Electrons supplied by the atoms are then

accommodated in the orbitals to achieve lowest overall energy

 Pauli’s principle + Hund’s maximum multiplicity rule

Hydrogen and He molecules

Hydrogen (H₂)

• Two electrons enter 1σ orbital

• Lower energy state than 2 H atoms

Helium (He)

• The shape is generally the same as H

• Two electrons enter 1σ orbital

• The next two electrons can enter 2σ* orbital

• Antibond is slightly higher energy than bonding

• He₂ is unstable than the individual atoms

Bond order

 A measure of the net bonding in a diatomic molecule

 n : number of electrons in bonding orbital

 n* : number of electrons in antibonding orbital

 Characteristics

 The greater the bond order, the shorter the bond

 The greater the bond order, the greater the bond strength

) 2 (

1 _*

n n

b = −

Period 2 diatomic molecules - σ ^orbital

 Elementary treatments : only the orbitals of valence shell are used to form molecular orbital

 Valence orbitals in period 2 : 2s and 2p

 σ-orbital : 2s and 2pz orbital (cylindrical symmetry)

 From an appropriate choice of c we can form four molecular orbital

z z

z

z A p B p B p

p A s

B s B s

A s

A c c c

c ₂

ψ

ψ

ψ

ψ

ψ

Period 2 diatomic molecules - σ ^orbital

 Because 2s and 2p orbitals have distinctly two different energies, they may be treated separately.

 Similar treatment can be used

 2s orbitals  1σ and 2σ*

 2p_z orbitals  3σ and 4σ*

s B s B s

A s

A c

c ₂

ψ

ψ

ψ

z z

z

z A p B p B p

p

A c

c ₂

ψ

ψ

ψ

Period 2 diatomic molecules - π ^orbital

 2p

, 2p

 perpendicular to intermolecular axis

 Overlap may be constructive or distructive

 Bonding or antibonding π orbital

 Two π_x orbitals + Two π_y ^orbitals

Period 2 diatomic molecules - π ^orbital

 The previous diagram is based on the assumption that 2s and 2pz orbitals contribute to completely different sets

 In fact, all four atomic orbitals

contribute joinlty to the four σ-orbital

 The order and magnitude of energies change :

The variation of orbital energies of

period 2 homonuclear diatomics

The overlap integral

 The extent to which two atomic orbitals on different atom overlaps : the overlap integral

 ψA is small and ψB is large  S is small

 ψA is large and ψB is small  S is small

 ψA and ψB are simultaneously large  S is large

• 1s with same nucleus  S=1

• 2s + 2p_x  S = 0



=

ψ ψ

τ

S _A^* _B

Structure of homonuclear diatomic molecule

 Nitrogen

1σ² 2σ^∗2 1π⁴ 3σ²

b = 0.5*(2+4+2-2) = 3 Lewis Structure

 Oxygen

1σ² 2σ^∗2 3σ² 1π⁴ 2π^2∗

b = 0.5*(2+2+4-2-2) = 2

last two electrons occupy different orbital : π_x and π_y (parallel spin)

 angular momentum (s=1)

 Paramagnetic

: N N

: ≡

Quiz

 What is spin-orbit coupling ?

 What is the difference between sigma and pi bond ?

 Explain symmetrical structure of methane molecule.

1 1

22 2

2s p_x p_y